Movement is one thing; the fast varying direction of traveling birds quite another. With old gear, dealing with motion was primarily a matter of sufficient depth of field/aperture setting, shutter speed, and determining a pre-focusing distance that birds might travel through. With manual lenses, pre-focusing distance of birds in flight can be sheer guesswork.
These days, modern auto-focus capability apparently manages this pretty well, though the time interval available for autofocus technology to determine the focusing distance can be fractions of a second. I've worked up a few details, as follows: Imagine, for example, at a distance of 80 feet, a bird is traveling 25 miles per hour toward the photographer at a 20 degree angle off the longitudinal axis of the lens. This pace of travel constitutes 37 feet per second, though the cosine of the angle translates into an effective travel speed of 34.5 inches per second. If we accept Nikon’s circle of confusion criteria (0.0013 inches), the depth of field for a 600mm at F4.0 at a focusing distance of 80 feet is 17.2 inches. At this distance, much of the depth of field is behind the bird in flight, leaving approximately 4.3 inches DOF in front. Once the shutter is electronically tripped, the camera and lens combination have a mere 0.12 seconds to determine the focusing distance, obtain proper focus by moving the relevant lens elements (perhaps a two elements doublet), and capture the image before the front edge of the bird is outside—i.e., in front of—the front boundary of the depth of field. Nikon—and certainly Canon and Sony also—use stepper and linear motor technologies which apparently provide sufficient focusing speed to handle this. Narrower apertures—or shorter lenses—make for a longer front-side DOF, lowering the focusing-speed constraint.
It seems to me that auto-focus speed is unrelated to pre-release and burst functionality.